There are just too many lordly characters in The Return of the King for Éomer to get all that much attention. Between Aragorn, Imrahil, Faramir, and his own uncle Théoden, there just isn't a ton of story space for Éomer, even if he does become King of Rohan over the course of the novel. We know he is a decent guy, and we know that he goes a bit nuts on the battlefield after Théoden is killed by the Lord of the Nazgûl. But that's pretty much the long and short of his plot involvement in this book.
The really interesting thing about Éomer is the way that he provides a foil for his sister, Éowyn. Clearly, they love each other, and Éomer is overjoyed when he finds out that Éowyn survived her battle with the Lord of the Nazgûl.
But the problem is, Éomer has everything Éowyn wants: glory, and the chance to win even more honor on the battlefield. Éowyn has to work much harder to get into the fight with Mordor than Éomer does because she's a woman, and Middle-earth is a deeply sexist place. Nonetheless, because Éowyn has to try a lot harder to get what she wants, she also receives a much bigger chunk of the novel's focus and attention than her brother does. Éomer fades into the background, but we remember Éowyn, which we are sure would be music to her ears.